--%>
Assignment Help - homework help

Solution Of Laplace’s Equation

1. Solve Laplace's equation for the electrical potential between two infinite parallel plates, which have a charge density per unit area -on one plate and a charge density per unit area -! on the second plate, and determine the electric field between the plates from the potential. (You should use the fact that for plates of infinite area, the potential only depends on distances perpendicular to the plates. (Since the object of this problem is to solve Laplace's equation for a problem for which you already know the electric field, if all that you do is to quote the value for the electric field that you already know from Gauss's law, without solving Laplace's equation, you will receive no credit.)

2. The Coulomb potential 1853_123.pngcan be written as r-1F(u, x), where F(u, x) = [1 - 2ux + u2]-1/2, where u = r!/r and x = cos Θ, where Θ is the angle between 1928_1234.pngand 1928_1234.png. Expand F(u, x) up to third order in u, and by doing so, demonstrate that the Taylor series for F(x, x), as a function of u,

2472_12341323.png

is valid to third order in u. (Therefore,

1810_54444.png

Incidentally, F(u, x) is known as the generating function for the spherical harmonics because its derivatives are proportional to the spherical harmonics.)

3. A point dipole is placed at the center of a grounded spherical conducting shell. Find the resulting potential inside the shell, (Hint: You must add to the expression for the potential due to a point dipole solutions to Laplace's equation that do not diverge at the center of the sphere, which we will assume to be the origin of coordinates.)

4. A long cylindrical conductor is placed in an initially (i.e., before the cylinder is present)uniform electric field which is perpendicular to the axis of the cylinder.

a) Find the electrical potential outside of the cylinder.

b) Find the surface charge density on the cylinder.

c) What is the electrical potential inside the cylinder when it is in the electric field.

   Related Questions in Physics

  • Q : What is Magnus effect Magnus effect :

    Magnus effect: The rotating cylinder in a moving fluid drags a few of the fluid about with it, in its direction of rotation. This raises the speed in that area, and therefore the pressure is lower. Therefore, there is a total force on the cylinder in

    		•
  • Q : Define Relativity principle Relativity

    Relativity principle: The principle, utilized by Einstein's relativity theories, that the laws of physics are similar, at least qualitatively, in all frames. That is, there is no frame which is better (or qualitatively any different) from any other. T

    		•
  • Q : Explain Daltons law of partial pressures

    Dalton's law of partial pressures (J. Dalton): The net pressure of a mixture of ideal gases is equivalent to the sum of the partial pressures of its components; which is the sum of the pressures which each component would exert when it were present al

    		•
  • Q : Define Copernican principle Copernican

    Copernican principle (N. Copernicus): The idea, recommended by Copernicus, that the Sun, not the Earth, is at the center of the earth. We now know that neither idea is accurate (that is, the Sun is not even situated at the center of o

    		•
  • Q : Define Weiss constant Weiss constant :

    Weiss constant: The characteristic constant dependent on the substance, employed in computing the susceptibility of the paramagnetic materials.

    		•
  • Q : Define Weber or SI unit of magnetic flux

    Weber: Wb (after W. Weber, 1804-1891): The derived SI unit of magnetic flux equivalent to the flux that, connecting a circuit of one turn, generates in it an electromotive force of 1 V as it is decreased to zero at a uniform rate in a period of 1 s; i

    		•
  • Q : Define Henry or SI unit of inductance

    Henry: H (after W. Henry, 1775-1836): The derived SI unit of inductance, stated as the inductance of a closed circuit in which the electromotive force of 1 V is generated whenever the electric current differs uniformly at a rate of 1

    		•
  • Q : Why tea kettle sing What is the reason

    What is the reason that the tea kettle sing? Briefly state the reason.

    		•
  • Q : What is Wave-particle duality

    Wave-particle duality: The principle of quantum mechanics that entails that light (and, certainly, all other subatomic particles) at times act similar to a wave, and sometime act similar to a particle, based on the experiment you are executing. For ex

    		•
  • Q : Explain Casimir effect Casimir effect

    Casimir effect (Casimir): The quantum mechanical effect, where two very big plates positioned close to each other will experience an attractive force, in the nonattendance of other forces. The cause is implicit particle-antiparticle p

    		•